1. 3.5 Domain of a Rational Function Thurs Oct 2 Do Now Find the domain of each function 1) 2)

2. Ch 1 Test Review Retakes: If you plan on retaking this test for 90%, see me at end of class Retakes must be scheduled for this week

3. Rational Function A rational function is a function f that is a quotient of two polynomials where q(x) is not the zero polynomial The domain of f(x) consists of all inputs x for which q(x) is not 0

4. Graphs of Rational Functions Various examples of graphs of rational functions can be found on page 301

5. Finding the domain To find the domain of a rational function, set the denominator equal to 0, and solve for x Note: any factors that you could cancel out still count towards the domain!

6. Ex Find the domain of each 1) 2) 3) 4) 5) 6)

7. Asymptotes An asymptote is a line that the function’s graph gets very close to but may not cross There are 3 types of asymptotes – Vertical asymptotes (x = ) – Horizontal asymptotes (y = ) – Oblique asymptotes (y = mx + b)

8. Vertical Asymptotes The line x = a is a vertical asymptote of the rational function p(x)/q(x) if: – X = a is a zero of the denominator – P(x) and q(x) have no common factors

9. Ex Determine the vertical asymptotes for the graph of

10. You try Find all vertical asymptotes for each function 1) 2) 3)

11. Closure Find the vertical asymptotes for HW: p.316 #7-13 odds, 69

12. 3.5 Horizontal and Oblique Asymptotes Mon Oct 6 Do Now Find the vertical asymptotes of

13. HW Review: p.316 #7-13 69

14. Horizontal Asymptotes The line y = b is considered a horizontal asymptote of p(x)/q(x) if: – As x approaches infinity, y approaches b – As x approaches neg. infinity, y approaches b Horizontal asymptotes only refer to a graph’s end behavior - A graph can cross horizontal asymptotes in the middle of the graph

15. Horizontal Asymptotes 3 cases: For each case you want to consider the highest power in the numerator and denominator – Case 1: Denominator’s power greater: y = 0 – Case 2: Numerator’s power greater: none – Case 3: Powers are equal: y = a/b where a and b are the lead coefficients of the num and denom

16. Ex 1 Find the horizontal asymptote

17. Ex 2 Find the horizontal asymptote of

18. Notes The graph of a rational function never crosses a vertical asymptote The graph of a rational function might cross a horizontal asymptote

19. Oblique Asymptotes A function has an oblique asymptote if the numerator’s power is exactly one higher than the denominator’s power To determine oblique asymptotes, we use long division Graphs can cross oblique asymptotes

20. Ex Find all asymptotes of

21. Ex Find all asymptotes of the function

22. Closure What is the difference between a horizontal and oblique asymptote? How do you find each one? HW: p.316 #1 3 5 15-25 odds

23. 3.5 Graphing Rational Functions Tues Oct 7 Do Now Find all asymptotes 1) 2)

24. HW Review: p.316 #1-5 15-25

25. Graphing Rational Functions 1) Find all asymptotes – Remember, can’t have both oblique and horizontal asymptotes 2) Find x and y intercepts – Plug in 0 for y and solve, for x and solve 3) For each region, test x-coordinates to determine where each curve occurs

26. Ex Graph

27. Ex2 Graph

28. Ex3 Graph

29. Closure Graph HW: p.317 #29-57 odds